The representation of rational numbers as sums of unit
fractions dates back to the time of ancient Egypt. Today this subject
survives mainly as a source of mathematical puzzles and problems in abstract
number theory, but the subject is also of historical and anthropological
interest, since it sheds light on the thought processes of people who lived
at the very beginning of recorded human history. The discussion below focuses
primarily on speculation as to the methods that may have been used by the
ancient Egyptians to construct their tables. Articles concerning the purely
number-theoretic aspects of unit fractions are contained in the Number Theory
section of this site.

1. The Rhind Papyrus 2/N Table

One of the most puzzling episodes in the history of human
thought is the 2000-year reign of Egyptian unit fractions. We can, at least
in part, reconstruct the arithmetical manipulations involved, but the
underlying reason or motive for expressing fractional quantities as sums of unit
fractions remains mysterious. Was it simply a cumbersome style of writing
that persisted for so many centuries just out of deference to traditional
forms, or did it express an actual way of thinking that has since been
forgotten?

At the beginning of almost every general history of
mathematics we find a description of how the ancient Egyptians operated with
fractions almost exclusively in terms of unit fractions. For example,
instead of saying 2/5 of my land was flooded, they would say 1/3 + 1/15 of my
land was flooded. One of the earliest written records from ancient Egypt
(transcribed circa 1650 BC from a source believed to date from around 1850 BC
or earlier) is known as the Rhind Mathematical Papyrus, and contains a table
expressing fractions of the form 2/n as sums of two, three, or four unit
fractions with distinct denominators. The table covers 2/n for n up to 101,
although the fractions with "even" denominators, e.g., 2/4, 2/6,
etc, are omitted, showing that they clearly perceived the obvious equivalence
of these with the reduced forms 1/2, 1/3, etc.

The first entry in the table is 2/3, to which they
assigned the expression 1/2 + 1/6. Every other table entry of the form 2/(3k)
is assigned the expression 1/(2k) + 1/(6k), which suggests they consciously
treated all denominators divisible by 3 as a single family, just as all
denominators divisible by 2 were implicitly treated as a single family.

Of the remaining table entries, the next is 2/5, to which
they assigned the expression 1/3 + 1/15. All but one of the remaining
denominators in the table that are divisible by 5 are assigned a simple
multiple of this expression, i.e., for 2/(5k) they used 1/(3k) + 1/(15k).
Similarly they assigned 1/4 + 1/28 to the table entry 2/7, and then "sieved
out" all the remaining denominators divisible by 7 using expressions of
the form 1/(4k) + 1/(28k). Finally, they assigned 1/6 + 1/66 to the table
entry 2/11 and then used 1/(6k) + 1/(66k) for 2/(11k) with k = 5.

The prime 11 seems to be where they stopped this procedure,
which is consistent with that fact that the table extended only to
denominators up to 101, so all the composites are sieved out by the primes
less than 11. It's remarkable that the Egyptians of 1850 BC (and probably
much earlier) had already developed this crude version of the "Sieve of
Eratosthenes", and seemed to have a grasp of the difference between
prime and composite numbers. Admittedly the sieve is not perfect, at least
not according to our present understanding. For one thing, the number 55 should
have been sieved out as a multiple of 5, but for some reason they chose to
treat it as a multiple of 11. Also, the composite numbers 35, 91, and 95 were
evidently not treated as composites, but were assigned unique
representations. Nevertheless, the overall impression is very strong that
they consciously sieved out the multiples of the smaller primes up to the
square root of the largest denominator in the table, and then treated the
remaining primes with unique representations.

As we've seen, for each of the small primes 3,5,7,11 the
Egyptians expressed 2/p as a sum of two unit fractions using the simple
formula

(The same formula also applies to the expression they
assigned to 2/23, although it may be coincidental.) Once these primes, and
their multiples, have been resolved, the table entries for the remaining
prime denominators suggest that the Egyptians determined

the representations by using the identity

where "a" is some convenient “round” number greater
than p/2. To find the remaining terms, we partition the quantity 2a - p into one, two, or three distinct parts
such that each part is a divisor of a. (This is why it's good to choose a “round”
number for a, so it has many divisors.) For example, with n = 89 we chose a =
60, which gives the difference 31. Thus, we need to express 31 as a sum of
three or fewer distinct integers each of which divides 60. One such partition
is 31 = 15 + 10 + 6, which leads to the representation that appears in the
Rhind Papyrus for 2/89:

On this basis, it's possible to summarize the 2/n table in
the Rhind Papyrus by giving the values of a,b,(c,(d)) for each prime p such
that 2/p = 1/a + 1/b + (1/c + (1/d)). These values are presented in the table
below.

TABLE 1: Summary of
Rhind Papyrus 2/n Representations

p 2a-p a b c d Also covers these

--- ------ --- --- --- --- -------------------

3 1 2 6 all multiples of 3

5 1 3 15 25, 65, 85

7 1 4 28 49, 77

11 1 6 66 55

23 1 12 276

13 3 8 52 104

17 7 12 51 68

19 5 12 76 114

31 9 20 124 155

37 11 24 111 296

41 7 24 246 328

47 13 30 141 470

53 7 30 318 795

59 13 36 236 531

67 13 40 335 536

71 9 40 568 710

97 15 56 679 776

29 19 24 58 174 232

43 41 42 86 129 301

61 19 40 244 488 610

73 47 60 219 292 365

79 41 60 237 316 790

83 37 60 332 415 498

89 31 60 356 534 890

exceptional
cases:

35 25 30 42

91 49 70 130

95 25 60 380 570

101 1111 606 101 202 303

This table raises two obvious questions. First, assuming
the Egyptians used something like formula (2) to determine their general unit
fraction representations for 2/p where p is a "large" prime, how
did they select the value of "a" and the partition of 2a - p from the available possibilities?
Remarkably, if we examine all the possibilities, and limit ourselves to just
the three and four-term representations where the smallest number x in the
partition of 2a-p is greater than 1,
then in most cases the expression appearing in the Rhind Papyrus is the one
for which a/x is minimized. For example, the only possible solutions for p =
43 are

partition of 2n-p

p a 2a-p x y z a/x

--- --- ----- ---- ---- ---- -----

43 24 5 2 3 12

43 28 13 2 4 7 14

43 30 17 2 15 15

43 30 17 2 5 10 15

43 36 29 2 9 18 18

43 42 41 6 14 21 7

and the representation appearing in the Rhind Papyrus is
the one with a/x = 7. In all, the Egyptians used the solution with the
minimum a/x for the "large" primes

13, 17, 19, 29, 31, 37, 41, 43,
59, 67, 73, 79, 83, 97

whereas they missed it for the primes

47, 53,
61, 71, 89

In these "missed" cases they missed the minimums
by 2, 6, 1, 3, and 1 respectively.

Another interesting fact that appears from a review of all
the possible representations for each prime is that p = 29 is the first prime
for which there is no three-term representation of 2/p (with the restrictions
noted above). Thus, it's not surprising that 2/29 is the first entry in the
Rhind Papyrus where a four-term representation is used.

The second major question raised by Table 1 is how to
explain the four exceptional cases. The first three are the composites 35,
91, and 95, that for some reason were not sieved out like the rest of the
composites. From our point of view the case 2/95 = 2/(5*19) should have been sieved
out by the small prime p = 5, giving it a representation of 1/(3k) + 1/(15k)
with k = 19. Instead, we find that its representation was evidently based on
the "large" prime p = 19, i.e., it is of the form 1/(12k) + 1/(76k)
+ 1/(114k) with k = 5.

The cases 2/35 and 2/91 are even more unusual, and in a
sense these are the most intriguing entries in the table. These are the only
two composites whose representations are not simple multiples of the
representations of one of their prime factors. Remarkably, in these two cases
it appears the Egyptians reverted from the normal multiplicative
decomposition to what might be called a "harmonic-arithmetic"
decomposition. Recall that the ancient Greeks had definitions for various
kinds of "means", including the

It's believed the Greeks inherited these definitions from
the Babylonians, but it's certainly possible they were also known to the
Egyptians. In particular, the harmonic mean certainly looks Egyptian,
given their affinity for unit fractions. In any case, notice that G(p,q) is
not only the geometric mean of p and q, it's also the geometric mean of
A(p,q) and H(p,q), which follows simply because

In other words, AH gives an alternative decomposition of
the composite number pq. This leads to the formula

where of course the leading factor on the right is a unit
fraction because p + q is even. This formula yields the Rhind Papyrus
representations

Thus we can say that every composite entry in the Rhind
Papyrus 2/n table is based on a decomposition of n into its prime factors. In
most cases the simple geometric factorization pq was used, but in two cases
they used the arithmetic-harmonic factors AH. (As to why the numbers 35, 91,
(and 95) might have been singled out for special treatment, see Appendix I.) This
leaves only the final entry in the 2/n table, which is

This entry could be constructed by formula (2) with a = 606
and the partition 1111 = 202 + 303 + 606, but it seems to stand out from the
other table entries due to the fact that it's a simple multiple of 1/n. Perhaps
this entry was just a formality, suggesting that for any n not covered in the
table (i.e., larger than 100), we can use the four-term expansion

so this effectively "completes" the table,
allowing us to say that it provides a unit fraction representation of 2/n for
all integers n. Interestingly, formula (4) can be seen as an
illustration of the "perfectness" of the number 6, in the sense
that the sum of the divisors equals double the number, i.e., 1 + 2 + 3 + 6 =
12 = (2)(6).

In summary, the 2/n table of the Rhind Papyrus, which
dates from more than a thousand years before Pythagoras, seems to show an
awareness of prime and composite numbers, a crude version of the "Sieve
of Eratosthenes", a knowledge of the arithmetic, geometric, and harmonic
means, and of the "perfectness" of the number 6. This all seems to
suggest a greater number-theoretic sophistication than is generally credited
to the ancient Egyptians. Whether they originated these ideas or borrowed
them, perhaps from the Babylonians, is unclear. (We shouldn't overlook the
possibility that the Babylonians borrowed them from the Egyptians.)

2. The Akhmin Papyrus

One relatively late document on Egyptian unit fractions is
known as the Akhmin Papyrus, apparently written around 400 AD. Considering
that the material in the Rhind Papyrus dates from 1850 BC (or earlier), this
shows that the use of unit fractions persisted for a remarkably long time. It
appears that by the time the Akhmin Papyrus was written there was a fairly
sophisticated criterion for the selection of the table entries. To expand
N/P, check the smallest solutions where exactly k denominators are divisible
by P using the congruences

k congruence modulo P

---- ------------------------------

1 Na = 1

2 Nab = a + b

3 Nabc = ab + ac + bc

4 Nabcd = abc + abd + acd + bcd

with 0 < a < b < c < d, and take the one with
the smallest maximum value. For example, to find the best expansions of n/17
we have the following choices for (a,b,c,d):

n k=1 k=2 k=3 k=4

--- ----- ----- ------- --------

2 (9) (3,4)* (2,5,6) (1,2,3,6)

3 (6) (4,5) (1,3,4)* (1,2,5,6)

4 (13) (3,8) (1,4,5)* (1,2,3,7)

5 (7) (2,4)* (2,3,5) (1,2,5,7)

6 (3)* (1,7) (1,2,4) (1,2,3,5)

7 (5) (1,3)* (3,4,7) (1,4,5,6)

8 (15) (1,5)* (2,4,6) (2,3,5,6)

9 (2)* (3,6) (3,4,5) (1,2,4,6)

10 (12) (1,2)* (1,3,6) (1,3,4,5)

11 (14) (3,7) (2,3,4)* (1,2,4,7)

12 (10) (2,6) (2,4,5) (1,2,3,4)*

13 (4)* (3,5) (1,2,6) (1,2,4,5)

14 (11) (1,4)* (1,3,5) (2,3,4,6)

15 (8) (2,3)* (1,2,7) (2,4,5,6)

16 (16) (2,5) (1,2,3)* (1,5,6,7)

The asterisks mark the solutions with the smallest maximum
term. The remarkable thing is that the asterisks also mark the expansions of
n/17 appearing in the Akhmin Papyrus. It's a perfect match. Clearly whoever
wrote that papyrus was organizing the solutions in

a way that is consistent with the method described here.

Applying this same analysis to the n/19 table in the
Akhmin Papyrus gives the results

n k=1 k=2 k=3 k=4

--- ----- ----- ------- --------

2 (10)+ (4,6)* (1,5,6) (1,2,3,6) <---

3 (13) (2,8) (3,4,5)*+ (1,3,6,7)

4 (5)+ (2,3)* (1,2,8) (1,3,4,5) <---

5 (4)+ (1,5) (1,2,3)* (2,3,5,6) <---

6 (16) (1,4)*+ (4,5,6) (2,3,4,6)

7 (11) (2,6)*+ (2,4,7) (1,4,5,6)

8 (12) (6,7) (2,3,5)*+ (1,2,4,7)

9 (17) (4,5) (2,3,4)*+ (1,2,3,5)

10 (2)*+ (3,6) (1,4,5) (1,2,3,4)

11 (7) (1,2)*+ (1,3,6) (1,3,4,7)

12 (8) (1,7) (2,4,6)*+ (1,2,5,6)

13 (3)*+ (1,8) (4,6,7) (2,3,4,5)

14 (15) (1,3)*+ (3,5,6) (1,4,6,7)

15 (14) (2,4)*+ (1,2,5) (1,3,5,6)

16 (6) (4,7) (1,2,4)*+ (1,3,4,6)

17 (9) (3,5)*+ (1,4,7) (4,5,6,7)

18 (18) (3,4)*+ (1,3,5) (1,3,7,8)

The expansion with the smallest max denominator is
indicated by an asterisk, and the one appearing in Akhmin is indicated by a
plus sign. In this case the match is nearly perfect, with just the following three
exceptions

fraction
Akhmin Expected

-------
--------- ------------------

2/19 10' 190' 12' 76' 114' (Rhind)

4/19 5' 95' 6' 38' 57' (2*Rhind)

5/19 4' 76' 6' 19' 38' 57' (2*Rhind+ 1/19)

In these three cases the Akhmin author selected the
expansion with the fewest terms, rather than the expansion with the smallest
max denominator. Interestingly, the "expected" series for 2/19 is
precisely the one that appears in the Rhind Papyrus, and of course the series
for 4/19 is just twice 2/19 (in Akhmin as well as in the expected series),
and the expected series for 5/19 is just 1/19 plus 4/19.

Even granting that the selection criterion for the Akhmin
tables was as described above, this still leaves the question of what algorithm
might have been used to compute the results. It occurs to me that the author
of the Akhmin Papyrus could have used a "meta-table" to construct
his tables. (Maybe he kept the meta-table secret for job security?) Basically
there are only a limited number of combinations of coefficients, so we could
build a meta-table just once and use it to construct all the individual n/p
tables.

Meta-Table For Ahkmin Papyrus Unit Fractions

A B a b c d A B a b c d

--- --- - - - - --- --- - - - -

2 1 2 5 1 5

2 3 2 1 20 9 5 4

3 1 3 15 8 5 3

6 5 3 2 10 7 5 2

3 4 3 1 5 6 5 1

6 11 3 2 1 60 47 5 4 3

4 1 4 40 38 5 4 2

12 7 4 3 20 29 5 4 1

8 6 4 2 30 31 5 3 2

4 5 4 1 15 23 5 3 1

24 26 4 3 2 10 17 5 2 1

12 19 4 3 1 120 154 5 4 3 2

8 14 4 2 1 60 107 5 4 3 1

24 50 4 3 2 1 30 61 5 3 2 1

40 78 5 4 2 1

To find the best unit fraction expansion of n/p, all we
need to do is take the first A,B from this table such that nA-B is a multiple
of p. For example, to expand 12/17 we try the first entry, A = 2,B = 1, which
gives 12A-B = 23, not a multiple of
17. So we try the next entry, A = 2,B = 3. This doesn't work either, so we go
on to the next. The first entry that works is the 14th: A = 24,B = 50.
Therefore, the optimum expansion of 12/17 is given by [a,b,c,d] = [4,3,2,1].

In this case the solution was given by the 14th entry in
the meta-table. If we check all the n/17 expansions in the Akhmin Papyrus we
find that the solutions are given by the meta-table entries listed below:

n -> 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

entry
# -> 8 12 22 9 3 5 19 1 2 11 14 7 10 4 6

Using this method the hardest expansion to find would be
4/17, because we have to check down to the 22nd entry in the meta-table (A = 20,B
= 29), but it isn't particularly laborious. With a little practice we could
probably do it in our heads. (Notice that we can take A and B modulo p, so
the 22nd entry with p = 17 is equivalent to A = 3, B = 12, and obviously 4(3)
- 12 = 0.) Actually to cover all of
the n/19 expansions they would have needed a meta-table going up to the 6's.
(I've just shown it up to the 5's.)

3. Why Unit Fractions?

Why did the ancient Egyptians (and others) persist in
their use of Egyptian fractions for so many centuries? Was it a conceptual
limitation, or simply a matter of notation? Some scholars contrast the exactness
of Egyptian expansions with the approximate nature of fixed-base expansions
such as in decimal system and the Babylonian sexigesimal system. This
contrast is interesting, although it takes some effort for modern readers
(accustomed to fixed-base representations) to imagine the intellectual
difficulties involved in this paradigm shift.

The Egyptian preference for exact expansions reminds me of
the Greek preference for geometry over symbolic arithmetic. When the Greeks
discovered irrational numbers they realized that rational arithmetic can only
approximate the values of most real numbers. As a result, not wanting to deal
with approximations, they devoted themselves mainly to geometry. Even within
geometry, their insistence on being able to give exact constructions
using straight-edge and compass is similar to the Egyptian insistence on
giving exact expansions using unit fractions.

It’s also interesting to compare the fixation on unit
fractions with the insistence of ancient astronomers on resolving the motions
of celestial objects into perfect circles – a practice that, like the use of
unit fractions, also persisted up to the 1500s. In both cases it seems to
have become a self-justifying end in itself. We resolve numbers into unit
fractions and we resolve motions into circles because those are
(respectively) the only “perfect” forms, even though the basis of these
concepts of “perfection” was not closely examined, and even though there was
no noticeable benefit in going through the contortions necessary to resolve
things into these (supposed) a prior forms.

There are at least two separate aspects to Egyptian
fraction expansions that makes them puzzling to modern people. One is the
variable "base", e.g., rather than expanding a fraction into a sum
of fractions with the denominators equal to powers of a single base number
(such as 10 or 60), they freely chose denominators to give an exact identity.
Thus, while the Babylonians might have expressed 1/7 as (approximately)

the Egyptians would have preferred the exact expansion

This also highlights the other puzzling aspect of Egyptian
fractions, namely, their preference for unit numerators. This might have
derived from their "binary" approach to integer arithmetic, in
which successive doublings of the operands were used to multiply numbers, so
that effectively their numbers were expressed in the form

where the coefficients ci are either 0 or 1.
When they expanded their arithmetic to include fractions, they might have
sought to express all numbers in the form

where again the coefficients ni are either 0 or
1, but realizing that using Dj = 2-j would not allow
exact expansions, they used independently variable denominators.

Still, it isn't clear what purpose was served by the
Egyptian unit fractions. Presumably one of the basic motivations for
expanding rational fractions is to enable the comparison of different
quantities. For example, if someone offers us 1/7 of a bushel of corn and
someone else offers us 13/89 of a bushel, which should we take? The
Babylonian approach would be to express the two numbers in sexigesimal as

This makes it easy to see that while the first terms are
identical, the second term of 13/89 is larger than the second term of 1/7. In
fact, it's clear that the fraction 13/89 exceeds 1/7 by about 6/602
+ 33/603. This shows the value of expressing fractions in a
fixed-base system: it enables us to immediately assess the relative
magnitudes of different quantities (and the difference between them) by
placing them on a common basis.

However, the Egyptian approach doesn't seem to serve this
purpose. One possible Egyptian expansion of 1/7 is 1/14 + 1/21 + 1/42, but
how would they expand 13/89? Using a "binary" approach, they might
have considered first expanding the numerator into powers of 2 as follows

Then from a table of 2/n expansions they would find

which immediately gives

The 1/89 term in the expansion of 8/89 could be combined
with the 1/89 in the original expansion to give 2/89, for which we could
substitute the 2/n table expression above. The terms 2/267 and 2/445 could be
written as (1/3)(2/89) and (1/5)(2/89) respectively, so again we could
substitute from the 2/n table expression. Adding up these terms would

give

This gives a complete unit fraction expansion for 13/89,
but it isn't obvious how this facilitates a comparison with 1/14 + 1/21 +
1/42. At some point, they would need to place the two numbers on a common
denominator.

Of course, we don't actually know how the ancient Egyptians
would have expanded 13/89, since the tables that have survived don't include
any general rules. Possibly they had some way of expanding the first few
terms on specified denominators for purposes of comparisons, but there is no
evidence of this. Based on the examples they gave, we would expect them to
expand 13/89 into something like

or perhaps, to minimize the largest denominator, the might
have used

but this still gives no easy basis of comparison with some
other fraction, such as 1/7 = 1/14 + 1/21 + 1/42. The most expedient way of
comparing the magnitudes would be to simply cross-multiply to clear the
fractions, finding that (13)(7) = 91 exceeds (1)(89) = 89, but again there is
no evidence the ancient Egyptians looked at it this way.

Although, there are undeniably several interesting
algebraic patterns in the historical Egyptian expansions, the purpose of those
expansions (i.e., the function they served) remains unclear. Were they just
exercises in manipulation, or did they serve some useful purpose? How did the
Egyptians compare the sizes of two general fractions? How did they add,
subtract, multiply, and divide general fractions? Did they use the 2/n
table for anything? Some scholars have suggested that the partitioning of
estates might have been one motivation, and it’s easy to see that this might
have given the Egyptians a special interest in unit fractions, but it’s not
clear what benefit they got from expressing unit fractions as sums of other
unit fractions.

It's interesting to consider other possibilities, such as
gambling. The modern theory of probability originated in a series of letters
between Fermat and Pascal on the subject of partitioning the stakes of an
unfinished game of chance. I'm no scholar of ancient cultures, but I'd be
willing to bet that the ancient Egyptians practiced some forms of gambling.
Maybe some forgotten predecessors of Fermat and Pascal were concerned about
the same thing, and worked out a set of mathematical techniques for dealing
with these kinds of partitions. Still, I can't quite see how to make use of
Egyptian unit fractions for any of these purposes.

One possible reason the practice of expressing numbers as
unit fractions endured for so long is the limitations of notation. Darrah
Chavey points out that the ancient Egyptians wrote a number 1/n as the number
n with an oval above it. This is just a single-variable symbol, and doesn't
readily accommodate the two variables needed to express the ratio of an arbitrary
numerator and denominator. It is necessary to devise a completely new notation.
Moreover, not only is a new notation required, it may have been difficult for
them to imagine a single quantity with two variable and independent
arguments. They could adjoin unit fractions by addition, but couldn't
conceptually consolidate them into a single entity.

Searching for clues to explain the motives behind the use
of unit fractions, we might examine the Rhind Papyrus itself. Recall that
Ahmes poses the problem of dividing 3 loaves of bread equally between 5 people.
Naturally each person gets 3/5 of a loaf, but there are multiple distinct
ways of partitioning the loaves to accomplish this. One way would be to cut
each loaf into five equal parts and give each person three parts. This would
require 12 cuts. Another way would be to make one cut in each loaf, dividing
it into 3/5 and 2/5 parts, and give each of three people one of the 3/5
parts. This leaves three parts of size 2/5. One of these could be cut in half,
and each of the remaining two people could be given a 2/5 and a 1/5 slice.
This would require only 4 cuts.

In the book "Ancient Puzzles", Dominic Alivastro
suggests that Ahmes might have wanted to solve this problem by cutting one
loaf into five equal slices, and the other two loaves each into three equal
slices. Then take one of the 1/3 loaf slices and cut it into five equal
slices. Each person could then be given his share in the form

Alivastro suggests that this might be more readily
perceived as equable than the partition 3/5, 3/5, 3/5, (1/5 + 2/5), (1/5 +
2/5), although it must be said that the uniform partition into congruent
shares (1/5 + 1/5 + 1/5) would presumably be even more obviously equable.
Both of these partitions requires 12 cuts, so we cannot prefer one over the
other based on economy of cuts.

Overall the most plausible explanation for the ancient
fixation on unit fractions seems to be that they had difficulty conceiving of
a single quantity in terms of two variables (numerator and denominator), and
were looking for simple "whole" fractional quantities. Just as the
"whole" natural numbers are those of the form n/1, it was natural
to imagine that the "whole" fractional numbers are of the form 1/n.

Appendix I: Why Were 35, 91, and 95 Treated
Differently?

It's intriguing to consider the following table:

Double-Triangular and Related Numbers

k T = (k+2)(k+3) Q = 6k+1 T-Q TQ

--- -------------- ---------- ------ -----

1 12* 7 5 35

2 20* 13 7 91

3 30* 19 11 209

4 42* [25] 17

5 56* 31 [25]

6 72 37 (35)

7 90 43 47

8 110 [49] 61

9 132 (55) (77)

10 156 61 (95)

11 182 67

12 210 73

13 240 (85)

14 272 (91)

15 306 97

16 342 103

18 420 109

Notice that the values of TQ less than 100 are precisely
those that are treated by arithmetic-harmonic decomposition in the 2/n table.
Also, I've placed parentheses around the composite values in the Q and T-Q columns, with square brackets to
indicate squares. Notice that the numbers 35, 91, and 95 appear, as do the
corresponding values of 2a-p, namely, the squares 25, 49, and 25
respectively. Also, the number 55 appears, which was treated in a slightly
unusual way in the 2/n table by being sieved out by the larger of its two
divisors, rather than the smaller. The only other composites in these columns
are 77 and 85, which don't seem to have been treated in any unusual way in
the 2/n table. By the way, the numbers in the T column, which are
double-triangular numbers, seem to have been favorite choices for
"a". Each value marked with asterisk was used in the 2/n table as
an "a" value at least once.